astronomy, calendar setting, and so on, for well over 1,000 years. Its accuracy was not challenged until the insistence on circular motion was abandoned and other curves were considered.
A model for planetary motion.
See: The Sun-Centered Universe, page 60; The Sun-Centered Universe Again, pages 64–65
628 AD India
Zero
Brahmagupta (598–665 AD)
The recognition that zero is a proper number.
_______________
Nowadays zero is an ordinary part of how we write numbers and calculate with them. It took a long time for it to be accepted as a proper number.
For most of recorded history the number system started with one. There was no number zero (0) when the system for dating years was devised in 525 AD. The system goes straight from 1 BC to 1 AD – there is no 0 AD.
There is no need for zero when using Roman numerals. The number 105 is CV (100 + 5) and we do not have to show there are no tens. By contrast, in a place value system, some indication is required.
Before modern times, arithmetic was done using a counting board, with one column for units, one column for 10s, one column for 100s, and so on. A symbol for zero isn’t required – just a space in the relevant column for the missing digit.
When writing numbers, it is useful to have a special symbol to show that a number is missing – such a symbol was used in Central America by indigenous peoples centuries ago. The Babylonians used two small wedges in their sexagesimal system.
Using a symbol for a missing digit in a place value system is not the same as recognizing zero as a number in its own right. That was invented by Indian mathematicians. In 628 AD Brahmagupta wrote out a set of rules for zero including:
• The sum of a zero and a positive number is positive.
• Zero divided by zero is zero.
Now we agree with the first but not the second. Division by zero is forbidden. If a car travels zero miles in zero hours, what is its speed? It is undefined.
9th century Iraq
Kitab wa al jabr wa al muqabalah (The Book of Shifting and Balancing)
Al-Khwarizmi (c. 780–c. 850 AD)
A comprehensive guide to solving quadratic equations.
_______________
Al-Khwarizmi’s book describes six types of quadratic equation and solves them in a methodical manner.
Kitab wa al jabr wa al muqabalah means: “The Book of Shifting and Balancing.” With an equation you shift terms from one side to another, and then you balance the terms collected on one side.
Say Al jabr quickly. This book is where the word algebra comes from. It is a long way from the algebra studied in schools today: for example, there is no use of letters such as x to stand for an unknown. Indeed, equations are written out in terms of words rather than symbols.
Al-Khwarizmi considers six separate types of quadratic equation, though nowadays the single quadratic formula covers them all. In his time there were no negative numbers, and so x2 – 5x = 6 had to be treated differently from x2 + 5x = 6.
The book contains no startling new discoveries. Its virtue consists in its systematic collection of different results and its methodical treatment of them, in an algebraic rather than a geometric way.
See: Quadratic Equations, pages 11–12
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